Commutativity in series of ordinals: a study of invariants

Abstract:

It is well known that two ordinals are additively commutative if and only if they are finite multiples of some given ordinal, and it is very easy to extend this result to any finite sequence of ordinals. However, no necessary and sufficient conditions for the commutativity of a series of ordinals seem to be known when the length of that series is infinite, although sufficient conditions for certain cases have been given by Sierpiński and Ginsburg. In this paper we present such necessary and sufficient conditions. The general problem is split into five distinct cases: those in which the length of the series is a regular initial ordinal, a singular initial ordinal, an infinite, noninitial prime component, an infinite successor ordinal, and an infinite limit ordinal that is not a prime component. These are dealt with respectively in the second through to the sixth sections of the paper, and it turns out that in every case our criteria can be expressed in terms of an ordinal parameter, which is in fact an invariant of the series in question. This concept of invariance is introduced in the first section, which also contains several lemmas and a slight strengthening of the original Sierpiński-Ginsburg result. The final section of this paper differs from the preceding four sections in two aspects. Firstly, the proofs of its two main results are merely sketched, since they contain no arguments that have not previously appeared in some form or other. Secondly, we have not given any explicit determination of the ordinal parameter introduced in this section, since we felt that such a determination would prolong the paper intolerably and encroach upon work done by J. A. H. Anderson: we have therefore simply referred to Anderson’s interesting paper.